Simple weak modules for some subalgebras of the Heisenberg vertex algebra and Whittaker vectors

Abstract

Let ${\mathscr M}(p)$ $(p=2,3,\ldots)$ be the singlet vertex operator algebra and $\omega$ its conformal vector. We classify the simple weak ${\mathscr M}(p)$-modules with a non-zero element $u$ such that for some integer $s\geq 2$, $\omega_i u\in{\mathbb C} u$ ($i=\lfloor s/2\rfloor+1,\lfloor s/2\rfloor+2,\ldots,s-1$), $\omega_{s} u\in{\mathbb C}^{\times} u$, and $\omega_i u=0$ for all $i>s$.